Syntax and Semantics of Quantitative Type Theory Robert Atkey University of Strathclyde [email protected] Abstract then this can easily lead to a runtime representation of vectors that We present Quantitative Type Theory, a Type Theory that records consumes memory space quadratic in the length of the list! usage information for each variable in a judgement, based on a Related to the problem of distinguishing between type formation previous system by McBride. The usage information is used to give and computational use of data is distinguishing between different a realizability semantics using a variant of Linear Combinatory kinds of computational use of data. Such information can be used Algebras, refining the usual realizability semantics of Type Theory to generate more efficient code, or to ensure that programs only use by accurately tracking resource behaviour. We define the semantics computational resources in restricted ways (allowing, for example, in terms of Quantitative Categories with Families, a novel extension in place update of memory). Linear Logic [14] initiated a body of of Categories with Families for modelling resource sensitive type research into such systems. Initially, this recorded zero, single, or theories. multiple uses (made explicit by Mogensen [28]), but recent work in coeffects and quantitative types by Petricek et al. [29], Brunel CCS Concepts • Theory of computation → Linear logic; Type et al.[8], and Ghica and Smith [13] refined this to track usage via theory; semirings. However, extending these systems to dependent Type Keywords Type Theory, Linear Logic Theory is not straightforward due to the conflict between type formation and computational uses. ACM Reference Format: McBride [25] has recently proposed a resolution to this conflict Robert Atkey. 2018. Syntax and Semantics of Quantitative Type Theory. In by combining the work on erasability and quantitative types. His LICS ’18: 33rd Annual ACM/IEEE Symposium on Logic in Computer Science, insight is to use the 0 of the semiring to represent information July 9–12, 2018, Oxford, United Kingdom. ACM, New York, NY, USA, 10 pages. https://doi.org/10.1145/3209108.3209189 that is erased at runtime, but is still available for use in types (i.e., extensionally). McBride presented a syntax and typing rules for the 1 Introduction system, as well as an erasure property that exploits the difference between “not used” and “used”, but does not do anything with the Dependent Type Theory promises to combine “programming” and finer usage information available. In this paper, we fix andextend “verification” by combining both in a single system. The implementa- McBride’s system, and present semantic interpretations that fully tions Agda [34] and Idris [7] advertise themselves as a “dependently exploit the usage information. typed functional programming language” and a “general purpose pure functional programming language with dependent types”, re- Contributions spectively. Coq [24] is primarily intended as a proof assistant, but 1. Section 2 reformulates McBride’s system as Quantitative also provides a program extraction facility. Type Theory (QTT) to add dependent tensor products and However, when trying to actually use Type Theory as a general booleans, and to fix a bug in the original system that caused purpose programming language, type dependency appears to ac- substitution to be inadmissible. tively encourage inefficient code. This is caused by Type Theory’s 2. Section 3 presents Quantitative Categories with Families, a use of variables for two purposes: as information to be used in novel class of categorical models for interpreting QTT. the formation of types, for example, the type Fin(n) of naturals 3. Section 4 presents several concrete realisability models of bounded by n depends on the natural number n when type check- QTT as instances of QCwFs. These demonstrate that QTT ing, but not a runtime; and as computational information that is allows resource sensitive interpretation of terms. Read con- manipulated by programs at runtime. An example is illustrated by structively, these interpretations yield an efficient extraction the type of the cons operation for length indexed vectors of Ss: mechanism with precise control over resource usage. cons : (n : nat) ! S ! vec n S ! vec (succ n) S 2 Quantitative Type Theory A naive implementation of length indexed vectors will store, for Combining dependency with linearity is not straightforward. We every element: a value s; a tail v; and a natural number n recording motivate McBride’s solution and our formulation of it. the length of v. If a unary representation of natural numbers is used, Permission to make digital or hard copies of all or part of this work for personal or 2.1 Dependency and Accountancy classroom use is granted without fee provided that copies are not made or distributed In Martin-Löf Type Theory, the term judgement has the form: for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the x : S ; x : S ;:::; x : S ` M : T author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or 1 1 2 2 n n republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. From Type Theory’s mixed computational/logical point of view, LICS ’18, July 9–12, 2018, Oxford, United Kingdom the context x1 : S1; x2 : S2;:::; xn : Sn has two uses. It describes © 2018 Copyright held by the owner/author(s). Publication rights licensed to Associa- the names used for resources which may be used by M to construct tion for Computing Machinery. ACM ISBN 978-1-4503-5583-4/18/07...$15.00 a resource of type T . It also describes the names used to refer to https://doi.org/10.1145/3209108.3209189 the extensional meanings represented by those resources used to LICS ’18, July 9–12, 2018, Oxford, United Kingdom Robert Atkey form the types S2;:::; Sn and T . This dual usage is illustrated in called “discharged assumptions” by Terui [33]). In all these systems, the judgement: information about how variables are used is recorded using a semir- n : Nat; x : Fin(n) ` x : Fin(n) (1) ing. Semirings are a natural structure to use: addition is used to sum up multiple uses of a variable, and multiplication is used to where Nat is the type of natural numbers, and Fin(n) is the type account for nested use. In the notation we will use in this paper, a of numbers less than n. The variables n and x play two different judgement in these systems looks like: roles: x is used as a reference to a resource that is transferred from ρ1 ρn the input to the output; and n is used to define the type of x. The x1 : S1;:::; xn : Sn ` M : T (2) fact that n is not used in the computation being described is not where the ρ ;:::; ρ are elements of the semiring indicating how explicitly recorded. 1 n the corresponding variable is used. In these systems, the zero of Linear Logic [14] uses presence or absence in a context to explic- the semiring is used to indicate that a variable is not used at all: itly record used resources. A linear typing judgement, 0 x : S is a complicated way of stating that x may as well be absent. ` x1 : X1;:::; xn : Xn M : Z However, when we move to dependent types, 0 usage variables indicates that the term M uses the resources named by the xi each have a useful meaning. McBride [25] reads the usage annotations precisely once. ρi as indications of computational usage, so a variable with usage Linear Logic’s resource accounting via presence conflicts with 0 indicates that it has no “run-time” presence, but may still be used type dependency. If a variable’s referenced resources are not used in the formation of types. The properties of semirings means that 0 computationally, it must not appear in the context and so is not usage is ideal for tracking use in types: we always have 0 + ρ = ρ, available for use in type formation. Judgement 1 is thus not linear: so combining a computational use with a use in a type retains the both because n is not used in the term, and because it is used twice original usage; and 0ρ = 0, so nesting an apparently computational in types. use within a type treats the whole usage as noncomputational. Thus To resolve this conflict, several authors have used formulations of McBride’s system incorporates both erasure (see also Miquel [26] Linear Logic that distinguish between unrestricted (“intuitionistic”) and Mishra-Linger and Sheard [27]) with linearity. variables and restricted (“linear”) variables. For simple types, this Term typing judgements now have the form: originates in Barber’ work [5], where judgements have two contexts ρ1 ρn σ separated by a vertical bar: x1 : S1;:::; xn : Sn ` M : T (3) x1 : S1;:::; xm : Sm j y1 : X1;:::;yn : Xn ` M : Y where the difference from Judgement 2 is that the output is anno- tated with a usage σ, where σ is restricted to either be the 0 or the 1 x The variables i may be used without restriction, zero, one, or of the semiring. This annotation means that we can construct terms y many times. The variables i must be used exactly once. Thus, that are only to be used in the formation of types. McBride allowed the judgement tracks information about the two different kinds of arbitrary usages ρ on the final colon. However, this yields a system usage. Cervesato and Pfenning [9], and later Krishnaswami et al.